 What does all this have to do with scalar geometry? What we want is to fit our problem of studying scalar Einstein metrics or similar things into this kind of general framework. And that can be done in two closely related ways. One is infodimensional and one by taking the limit of fine dimensional pictures. But so the infodimensional approach, the relevant group is the G, it's the symplectic diffeomorphism group. So we have a scalar manifold, X, but we can just forget about the complex structure and just think about the symplectic manifold and all the symplectic forms under consideration will be equivalent as symplectic forms so it doesn't matter which one we choose. Probably more precisely we should call it the Hamiltonian diffeomorphism group or something, but that's not going to be important. And we can think of this as acting, so A will be the set of almost complex structures, compatible almost complex structures. We fix this symplectic form and then we can look at this thing and you can write down in a formal way, these are infodimensional spaces, but you can write down in a formal way the structure that we had, a symplectic form and metric, an infodimensional scalar structure on this and so forth. Then the moment map is just given by the scalar curvature. So mu of J is essentially the scalar curvature of the metric determined by a complex structure and the symplectic form. Again, this is a slight distortion of the truth, but it's true for our purposes. Actually minus the average value of the scalar curvature. Just subtract off a constant. So this is a function which is essentially in the dual of the Lie algebra. The Lie algebra of this is the function, so this is the right kind of thing to be, the dual of the Lie algebra. So the things that we are looking for, the things that we expect to meet in this general picture are just the constant scalar curvature metrics. I'm saying a zero at the moment map will be something with constant scalar curvature where the scalar curvature is equal to its average value. So in fact, I don't have an almost complex structure, but we're always just going to restrict to the subset of integrable complex structures. In fact, we're going to really restrict to a single orbit in our discussion most of the time, so you can worry about that. So you can write down a lot of what we had. What you can't really write down is the complexified group because that doesn't exist in any genuine sense as a group, but you can still make enough sense of it to carry through essentially everything we've said. In particular, we can interpret the quotient space Gc over G as the space of scalar metrics. If we fix one complex structure, we can interpret this by thinking of this as an orbit or the quotient by G of an orbit in the almost complex structures. We can think of this as a familiar space of all scalar metrics in a given class, i as a sort of omega naught plus i d bar d5 for some fixed scalar class. So the symmetric space structure we expect on this does more or less make sense. It corresponds to writing down a Romanian metric introduced by Mabucci, which is just that if we take a tangent vector, we're at a given point, we take a tangent vector is just given by a change in phi, so we want to say what the norm of delta phi is, but it's just the obvious thing. We take the, let's call this thing omega phi, we take the L2 norm with respect to the usual measure determined by this metric. So the only interesting thing which is happening is that this metric measure is varying as we vary over our space. So let's call this d mu phi. We want to make sense of the geodesics in this space in some form, and these are functions of phi t, geodesic. If you can write it all out, what you find is that these correspond to solutions of the degenerate, the homogeneous Bonjour-Jampere equation on a larger space. So if you think of taking x times s1 times an interval, if our t is in our interval, when we think of this as a scalar potential on this space, but the circle acting trivially, then we just corresponds to a form which is null. So solving the geodesic equation just corresponds to solving a, so this homogeneous, this is essentially d, if I think of phi as the function of two variables corresponding to this one parameter family, essentially we're looking at the equation d bar d of phi to the m plus 1 is equal to zero. And we can write down what this functional is corresponding to the norm. So it's called, again, goes back to Mabuchi, Mabuchi functional, which is, again, essentially we define, what we define is its derivative. So you need to see that this actually, so we're really defining a one form on our space. You need to see it's a closed one form, so up to a constant defines a function, but that's the case. So the formula, again, is that the change in n, infinitesimally as we change by phi, we just take this scalar curvature minus the average value of the scalar curvature. D mu of phi. Does this notation make sense to you? So as we see, the critical points, I'll just say the scalar curvature is constant, as we say. And then there are other things. Another basic important functional that comes in is just defined by, would I, is just defined by taking the integral of delta phi. So this is kind of a formal setup for understanding the constant scalar curvature equation on the scalar manifold. In fact, if we're working in the right topological setting, if we're working in, if the scalar class is the first-gen class of the manifold, then any constant scalar curvature metric will be scalar Einstein. So I have a certain identity. So we can think of this as a way of understanding the scalar Einstein equations also. But in fact, there's a, that's a rather, since it's embedding a problem in a harder problem in a sense, it's embedding a second-order equation, a fourth-order equation. There's a more direct, but to me, more mysterious way of setting up the scalar Einstein equations as functional by this, it's called the ding functional. Well, let's see, no. Another functional one can define, which is equal to, we take this, this I, and then we take minus the logarithm of the integral of, which is omega phi, so it works. So this isn't the case when the line bundle we're considering over our manifold is equal to the dual of the canonical bundle. So if we have a metric, so phi is to giving us a metric on the line bundle in a more invariant sense. So if we have a metric on this, it just corresponds to a volume form. And that's what this is. So, more of an incoordinance, this is essentially e to the phi, up to a constant factor. So this is something we can only define in the scalar Einstein setting, but again, the critical points are the scalar Einstein metrics. And remarkably, this is a result of Bo Vanson. This is also convex along gd6. I don't, there's no way that's true, but it's a fact, this is also convex. All three of these are convex? All three of these are convex. Well, this is linear in fact on gd6. This is convex by the general theory more or less. This is convex by a hard calculation. I don't know, kind of a... And that's something that actually comes into the story in an important way, at a technical level later. In the sense that this geodesic equation is not an elliptic equation. One can try to understand it in a certain sense, but solutions will generally have singularities. So if you have singularities along your... If you're trying to understand these functions along a geodesic and you have singularities, it's not quite clear how it's going to work, because what does the scalar curvature mean if you have a very singular thing? But this thing is much... Of course it doesn't depend upon semi-derivatives. So Bo is the expert side, but roughly one can easily produce kind of weak solutions of this equation. And the advantage of this being functional is it's still defined along such... You don't really care about the singularities. Okay, so this is something about the kind of inferred-dimensional picture. So because it's inferred-dimensional, hard work is required often to actually make the kind of formal properties work. It gives a context for thinking about things. So another approach is by taking the asymptotics of fine-dimensional set-up. So here we just consider... We just think of varieties in projected... Some fixed-projective space. So think of this as a projected space with its fixed Froobini-Study metric. We think of, speaking of the set of all varieties of a given numerical invariance of, let's say, the chow variety of such a thing. And then we have... Our group G would be SUn plus one. That's on the chow variety. And also the complexification, just so that this is just... And in fact, this chow variety can be embedded in some big projected space. So we can reduce to the standard picture if one likes. Any case, there's an actual metric and a line bundle and so forth on this chow variety. And what you find is that the moment nap of some variety V is just given by... So this should be a skewer joint matrix. And the layout of it here is just given by taking the integral over V. We take this formula. So I'm thinking of Z-alpha, the standard coordinates on our projected space. If I make this combination, it's homogeneous. So it's a well-defined function of projected space. This just means we take the trace-free part when we project out to get to them. This is a momentum after the action on the chow variety. That's the momentum after the action. Another way of saying that is if we take a fixed... Let's focus on a given one parameter subgroup defined by some A, then mu of V, A is equal to the integral over V of HA where this HA is the standard Hamiltonian on projected space for the corresponding action on projected space for the one parameter subgroup. I'm probably... You put some I's in somewhere to make this work. So it's a very natural thing. So let's... We want to understand this numerical invariant. When we have something fixed by one parameter subgroup, we look at the weight of the action on the fiber. That's just given by... Well, essentially by this formula. But let's... When we take account of this projection to the algebra, what we can write down is the following formula that we call the chow invariant. So supposing V is fixed by X, A, T the one parameter subgroup, then the chow invariant is going to be... We'll write this with the trace of A over the dimension, which is N plus one, minus one over the volume of V times the integral of HA. So this is essentially the same formula all we've done has been normalized it If A is a multiple of the identity, this vanishes. It's just a normalized function. That's this numerical invariant that we wrote down. The weight of the action. The thing we call the I of X lambda. Something like that. We go to the fixed point and we write down this weight of the action. This is it. So the condition that the chow... So what branches is the condition of the stability, the chow stability, it would be called, would be that this chow is positive. So what we're saying is that we start with some... Maybe we've got a class of notation. Let's call it V naught, say, fixed V naught. We start with some V. We want to understand if it's chow stable. What that means is that we take any one parameter subgroup and we flow it until we flow it into something fixed. So we get some V naught. And we want to say, in a stable situation, if this chow applied to the fixed thing is positive. That's quite a natural meaning. What that's saying is that to the trace of... This is just the average value of H on projected space. This is the average value of H on V naught. So the stability condition is when we flow downwards, which is going to be very complexified, we're taking the gradient flow of H. When we flow under the gradient flow of H downwards, the average value in the limit should be less than the average value on the ambient space. So you sort of feel... Unless it gets hung up in some bizarre way, that should be true. What's that saying? The integral of H... What's that, the integral over? So this thing is the average value of H... Of the projected space. The next integral is V naught. This is V naught. So this is the average value of a V naught. So this is something we could write down for a general... In Iranian context, we say we're looking at sub-manifolds that when you flow down, the average value in the limit is less than the average value on the ambient space. So this is a notion of Chao stability for something in a fixed projected space, which is quite a natural thing or something we can understand. But we want to think about the asymptotics of that. So now we're going to... The notation's going a bit weird. So now going back to the picture of an X with an L, for each K, we can embed X in some... If K is sufficiently large, we can embed by the linear system of sections of this thing, the projected space. And then we can apply this picture. On the other hand, if I have a... Supposing V naught is... Supposing we take K equals 1 and V naught appears there, then we can take the sections of... We can re-imbed it in a bigger space by taking the powers of the sections. So we can get the same V naught. We can also have this line bundler L on it. So rather than having a single... So now we would like AK to be the generator of the action on H naught of V naught L to the K. Changing my notation. Yep. So we have a V naught, this line bundle over it. This automorphism acts on all these spaces. And so we get this. And so we can write down this expression not for all possible Ks. So we can... Trace AK over... This is NK, we're calling it NK plus 1. Minus... We'll take the same... This... This sort of hammer... This integral actually won't vary with K. We normalize up to some normalization. This is independent of K. Integral. So maybe I should put a K. I put a K in here to make it exactly right. V zero is going to depend on K, right? If V zero is going to depend on K, because the chal variety you're going to be in is going to depend on K. What I'm doing is I'm slightly switching between everything. Once we've got a V zero, we've got a line bundle over it. Now we can take the powers of the line bundle to get a whole... We can bet it in a different projector space. If you're looking at chal varieties for different projector space... Then it'll be in a different... Yeah, that's right. It's like all the different V zero in some sense. It's the same V zero, but we're embedding it in bigger and bigger projector spaces. Yeah, that's right. So I'm slightly... Yeah. So we can write down this thing. But these... At least when K is large, these things have... These are given by Hilbert polynomials. These are polynomial functions. And where we've normalized it, this thing will have a limit as K tends to infinity just by general... This will be a polynomial of degree n plus 1 in K, and this will be a polynomial of degree n in K. So we've got the... And... Um... Yeah. For the way we've set it up, the top terms cancel, so the next term is a constant, and this is the futaki invariant of... Well... Maybe we could call it V naught a, or e to the a t, or something. V naught with this action, essentially. So might it happen that the quantity in the parentheses is negative for some low values of K in some of the states? Oh, yes. Yeah, we don't... This could jump her up. Yeah. So, and in fact, when we go to these asymptotics, there's some subtleties. So things that ought to be the same at first sight actually could be a bit different, but... But anyway, this is a correct... This is a formula that defines a certain numerical invariant. And that's what we're going to use to define stability of having... So in a sense, we could have cut out the last 40 minutes and just written down the definition. The last 40 minutes is being given. So why are we defining it in the way we're defining it? We need another lecture then, I think. I think we need another lecture. We're going to get one more. What time? What time is it going to be? Tomorrow. Tomorrow. We'll see. I mean, it could be at 12 or something like that. We can discuss what time. But you're right, I'm not going to... I think it's worth explaining that rather than just writing down that. Anyway, so now we're going to define the definition of case stability. So when we're talking about we start, we take a v and we move it by one prime of the subgroup to get a v0. That's actually the same as talking about... that's equivalent of talking about a family over c. Which is called a flat family or something like that. So I'm going to change my notation. So this is supposed to be c star equivariant. So as the c star acts on c in the usual way, we're supposed to have a lift of the action but we're supposed to have pi minus 1 of t is isomorphic to x for non-zero t. But pi minus 1 of 0, we should think of as being something different, x0. We're going back to about x's rather than v's. The same thing. And we also have line bundles on top of everything, but let's not write that down. The the futarkian variant, changing notation of this gadget is just the futarkian variant of x0 with the action. The c star action gives an action on the fiber over 0. Anything we define there. So this is a good algebra geometric definition because it doesn't really matter if this makes sense even if this is a scheme or whatever. We already just know that these things are given by Hilbert polynomials to get a good definition. What does it have to do with our other picture, our kind of input dimensional picture, the differential geometry that at least if if x0 is let's say smooth initially or not too angular then we can write down a formula essentially what we've written down before. It's given by taking the Hamiltonian for the action and pairing it with the scalar curvature minus the average value. So this is essentially a version of the Riemann-Roch theorem in a sense. There are a kind of Riemann-Roch formulae for this that expresses this in terms of carmology and you write that in terms of differential forms and you get such a formula. So this is if x is sufficiently smooth. H is the Hamiltonian for the induced action on this central fiber. A is the action on the Yeah, exactly. It's what we call H. I think I don't understand it. You had this C star action but over here you had like an ambient projector space. So you had this ambient unitary group but here we just have a sort of abstract family of varieties. Do you put them all in some big projector space or something? Yeah, you can pass between the two pictures. When you have this you take the direct image of the H naught of L to the K and you trivialize that here and then you put them all in a big projector space. One parameter group sits in the GLA and then you're over in the other picture. Yeah. Essentially, yeah, certainly. So yesterday you You need to understand equivariant bundles, vector bundles over C but you do. So as you say this is the definite. So the K stability, let's not write it as we've written, it says that if we have a given X you should have a fixed line bundle as well, is stable if any time you fit it into a family like this, generation test configuration, it's called this futaki invariant is positive. With some some extra few words to cover borderline cases but that's the point. It's the case to build an X and we can put X in this family. Then that should be positive. But there is a kind of infidimensional words, the related infidimensional discussion, in which you can translate these one parameter subgroups into geodesic rays that's what you should expect. It's an infidimensional notion where you say you look at geodesic rays and you want you can define the futaki invariant that way essentially from the limit of the derivative of this babuchi functionalism. What does geodesic rays correspond to points at infinity? You can compact it by a symmetric space? Well that's the kind of thing one right but that to a certain extent people have made sense of this and proved that if you have this algebra-geometric picture you can get in a sort of a weak sense of a geodesic ray. So roughly we should think of all these algebraic degenerations as you say as points at infinity in this space. Impacted by symmetric spaces by looking at equivalence classes of geodesic rays? Right. But I doubt the end of that works on the nose because it's all but it at least works enough to see that you have this is the right definition to be working with. So this is essentially what we're writing down is very close to what was defined by Tiar in the 1990s. At a technical level you get different definitions depending on just what kind of singularities you allow in your x-naughts. So let me just mention that in fact we can set things up we can set the definitions up a bit differently in a way that we don't need this c-star action. And at the end of the day you get the same concept. So let me just explain how they're supposing you just know you have a family x over say over a disc, say, no with pi-1 of T isomorphic to x or fixed x for a non-zero T. Then rather than thinking about the weights of this action and so forth what we've been doing, you can also think about there's a certain both for the Chow and for the Futaki story, they correspond to certain lines that you can functionally correspond to a variety. There's a certain way you have a variety, you write down a line associated to it. So if you believe that then given this family we get a line bundle over the disc not very interesting, but this isomorphism means that the line bundle is trivialized away from zero. So we have a relative germ class in variant and a line bundle on delta trivialized over delta star simplifies to green. And that's the same as the other thing that we defined. It can be formed from those or from this this Deline construction of a way of pushing down line bundles. It's not various ways of writing it. Okay, so finally I think we can get on to the his metrics with cone singularities. So it'll fit in best the way things are turning out to postpone any kind of analysis. I was going to talk about the linear elliptic theory on these a bit, but let's postpone that to the next lecture. Let's just see how we can fit into this sort of futaki invariant storing. So what we're considering of course is the model cone with angle beta is given by this metric in polar coordinates like a cone angle 2 pi beta. So we're going to say something more precise about this, we're going to consider some D in X more precisely in the linear system minus mu times K for some fixed mu positive mu and we're going to consider calometrics which have got a singularity modelled on this transverse to D. But let's not go into any kind of more detail, but let's just say that we can think in a certain natural sense which could be made rigorous, we can think of the such a metric the curvature, for example the scalar curvature having a delta function contribution along D. This cone you can think of in the two-dimensional picture is having Gauss curvature a delta function at the origin. So now we want to define a modified phutarchian variant. In fact, we can modify the whole story of all this stuff to put these divisor in, but the crucial thing is the phutarchian variant of change notation, I seem to have got let's call it, let's suppose we can change to fit in what we're going to do later, let's take Z rather than X and delta rather than D the phutarchian variant with this parameter beta of this pair when we have a C star action on this setup preserving so what we can let's write down the formula, this is the ordinary phutarchian variant as we defined before plus plus a correction term essentially is the integral of the Hamiltonian over delta but we'll put some of the normalizing factor in as well so this is a definition and again this we can also express this algebra geometrically if we want in terms of by writing these as the leading term in Hilbert polynomials why is this for good definition if we take our previous formula for the phutarchian variant but now we substitute in the, so we compute in the scalar curvature but now the scalar curve should have this delta function term on D but now that's precisely this or minus this so we're just subtracting off the delta function term so this is equal to let's write as the integral over z minus delta to emphasize of s minus s hat I the scalar curvature not including the delta function contribution this is what I'm inviting this integral so this has precisely got the property that this vanishes if we have a metric of constant scalar curvature with this delta function singularity this thing will be zero so I can let's go back now finally to discuss the strategy as I as I pretty much repeat what I said at the end of my talk on Friday but now we have more background to understand it so this is we for reasons I won't go into there is a scalar Einstein for small beta there is a there is one of these scalar Einstein metrics for some values of beta doing a small and also the set of the set where you have a solution is open so you can deform things so what we want to know if it's starting from some small value where we have this metric with a cone singularity we want to deform it all the way up to beta equals 1 so so the crucial thing is the closeness see we want to say if we have some sequence beta i increasing to a limit beta infinity and we have solutions x actually x i will be x but it's more of the integral x i solutions what we want to do we want to be able to say we can do for these singular metrics just what we did for the ordinary metrics in the first talk yesterday so we want to have some limit we want to have a Gromov-Hausdorff limit which has actually got an algebraic metric structure and we're going to have carrier on we have these d i's as well in the natural way we said done sir i said the words we want to have a limit which is an algebraic metric take an algebraic variety super kind secondly we want to fit one of these c star equivariant families such that such that the central fiber is z but of course we'll have the divisor is what we're going to have the family with the original d and the divisor deltas in the obvious way and secondly we want to be able to say that the futaki invariant for this beta infinity of x is 0 that's what I'm going to say but those are the three things we want to establish about that because if we've done that then we've proved our theorem essentially because this futaki invariant of beta is just linear we didn't write it down it just varies linearly with beta so by the easy directions normally called in this theory when you do have a solution the thing was stable so the futaki invariant back here was positive there are various ways of proving that various proofs in the literature it was 0 here so when we went out to 1 which is when it's the ordinary futaki invariant it has to be negative of course there's a special case when beta infinity is 1 and then you have to get the idea that's a good place to stop more or less I haven't got past I hope I've explained these are actually technical things to do if you're not coming tomorrow you should believe this is a variant it needs more work but it's a variant of what we did before it's the same idea this is this requires some technical words and quite a lot of technical work but it's not too hard to believe formally you'd expect this because we're also going to have a limit of our metrics in some sense here which is going to have a singularity along delta in some sense and if everything was sufficiently smooth then the formula we wrote this would be a constant scalar curvature metric so if we could apply the differential geometry formula this thing would be true so what the work involves again involves deep work but what it involves is saying that the singularities are not too bad that the differential geometry still works interesting